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Alexander Belavin

Publications and source records attributed to Alexander Belavin.

At least 19 recordsLinked to original sources
arXiv2026

Hodge numbers for orbifolds of Calabi-Yau threefolds Fermat type and the Roan pairs

First, for orbifolds of Calabi-Yau threefolds of Fermat type, we define Roan's Hodge numbers. We prove, Theorem \ref{main} , that for all orbifols of Calabi-Yau threefolds Fermat type, Roan's Hodge numbers correctly count the stringy Euler numbers due to Vafa formula. Second, for Calabi-Yau threefolds Fermat type, we apply Roan's Hodge numbers to get a relation between the Borcea-Voisin construction \cite{Borcea, Voisin} and Berglund-H\"ubsch-Krawits mirror symmetry (BHK mirror symmetry). Third, for orbifolds of K3 surfaces Fermat type, we establish relations twisted and untwisted parts of the second mixed cohomology to deformations and the Roan pairs. This allow us to confirm the BHK mirror symmetry for such orbifolds, since the Euler numbers computed by the Vafa formula are equal $24$ for all of them.

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arXiv2026

Free-Field Construction of Heterotic String Compactified on Calabi-Yau Orbifolds via Correspondence with $\mathcal{N}{=}2$ SCFT Minimal Models

We establish a correspondence between the free-field construction and the minimal-model construction of the Calabi--Yau sector of the four-dimensional heterotic string compactified on Berglund--H\"{u}bsch type Calabi--Yau manifolds and their orbifolds. For Fermat-type polynomials the Calabi--Yau vertex operators expressed in terms of free fields are shown to correspond to products of primary fields of $\mathcal{N}{=}2$ minimal models. Using this correspondence we verify modular invariance of the free-field construction and extend it to Berglund--H\"{u}bsch Calabi--Yau orbifolds, deriving the conditions on complete vertex operators that parallel those of the minimal-model construction.

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arXiv2026

Construction of mirror pairs Calabi-Yau orbifolds of the Berglund-Hubsch type

In this paper we have developed general algorithm for finding all orbifolds of Berglund-Hubsch-type Calabi-Yau manifolds and their mirrors. An explicit construction is formulated for finding all admissible deformations and groups defining mirror pairs of orbifolds. Then using our algorithm for one of the Calabi-Yau manifolds, defined by a Fermat-type polynomial, we found all mirror pairs of orbifolds. For this model, for each pair of orbifolds, the number of generations and the number of singlets i.e. particles participating only in gravitational interactions (dark matter particles) were found.

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arXiv2025

Free field construction of Heterotic string compactified on Calabi-Yau manifolds of Berglund-Hubsch type in the Batyrev-Borisov combinatorial approach

Heterotic string models in $4$-dimensions are the hybrid theories of a left-moving $N=1$ fermionic string whose additional $6$-dimensions are compactified on a $N=2$ SCFT theory with the central charge $9$, and a right-moving bosonic string, whose additional dimensions are also compactified on $N=2$ SCFT theory with the central charge $9$, and the remaining $13$ dimensions compactified on the torus of $E(8)\times SO(10)$ Lie algebra. The important class of exactly solvable Heterotic string models considered earlier by D. Gepner corresponds to the products of $N=2$ minimal models with the total central charge $c=9$. These models are known to describe Heterotic string models compactified on Calabi-Yau manifolds, which belong a special subclass of general CY manifolds of Berglund-Hubsch type. We generalize this construction to all cases of compactifications on Calabi-Yau manifolds of general Berglund-Hubsch type, using Batyrev-Borisov combinatorial approach. In particular, starting from the mirror pair of Batyrev lattices corresponding to a given CY manifold, we construct vertex operators of the complete physical theory as cohomology of Borisov differentials that correspond to points of reflexive Batyrev polyhedra. In particular, we show how the number of $27$, $\overline{27}$ and Singlet representations of $E(6)$ is determined by the data of reflexive Batyrev polytope that determines this CY-manifold.

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arXiv2024

Conformal bootstrap and Heterotic string Gepner models

In Gepner's pioneering work, the requirement that leads to a model having the desired $N=1$ Spacetime supersymmetry and $E(8)\times E(6)$ Gauge symmetry was the requirement that the spacetime symmetry is compatible with modular invariance. In this work we show that the requirement for the simultaneous fulfillment of mutual locality of the left-moving vertices of physical states with the space-time symmetry generators and of right-moving vertices with generators of $E(8)\times E(6)$-gauge symmetry, which arises after some special reduction together with the requirement of mutual locality of complete (left-right) vertices of physical states among themselves leads to the same Gepner models.

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arXiv2023

Mirror symmetry and new approach to constructing orbifolds of Gepner models

Motivated by the principles of the conformal bootstrap, primarily the principle of Locality, simultaneously with the requirement of space-time supersymmetry, we reconsider constructions of compactified superstring models. Starting from requirements of space-time supersymmetry and mutual locality, we construct a complete set of physical fields of orbifolds of Gepner models. To technically implement this, we use spectral flow generators to construct all physical fields from the chiral primary fields. The set of these spectral flow operators forms a so-called admissible group $G_{adm}$, which defines a given orbifold. The action of these operators produces a collection of physical fields consistent with the action of supersymmetry generators. The selection of mutually local fields from this collection is carried out using the mirror group $G^*_{adm}$. The permutation of $G_{adm}$ and $G^*_{adm}$ replaces the original orbifold with a mirror one that satisfies the same conditions as the original one. This also implies that the resulting model is modular invariant.

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arXiv2022

Explicit construction of $N = 2$ SCFT orbifold models. Spectral flow and mutual locality

In this work we present a new approach to constructing Calabi-Yau orbifold models required for compactification in superstring theory. We use the connection of CY orbifolds with the class of exactly solvable N=2 SCFT models to explicitly construct a complete set of fields in these models using the twisting of the spectral flow and the requirement of mutual locality of the fields.

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arXiv2020

Periods of the multiple Berglund-Huebsch-Krawitz mirrors

We consider the multiple Calaby-Yau (CY) mirror phenomenon which appears in Berglund-H\"ubsch-Krawitz (BHK) mirror symmetry. We show that for any pair of Calabi--Yau orbifolds that are BHK mirrors of a loop--chain type pair of Calabi--Yau manifolds in the same weighted projective space the periods of the holomorphic nonvanishing form coincide.

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arXiv2020

On the equivalence of Batyrev and BHK Mirror symmetry constructions

We consider the connection between two constructions of the mirror partner for the Calabi-Yau orbifold. This orbifold is defined as a quotient by some suitable subgroup $G$ of the phase symmetries of the hypersurface $ X_M $ in the weighted projective space, cut out by a quasi-homogeneous polynomial $W_M$. The first, Berglund-H\"ubsch-Krawitz (BHK) construction, uses another weighted projective space and the quotient of a new hypersurface $X_{M^T}$ inside it by some dual group $G^T$. In the second, Batyrev construction, the mirror partner is constructed as a hypersurface in the toric variety defined by the reflexive polytope dual to the polytope associated with the original Calabi-Yau orbifold. We give a simple evidence of the equivalence of these two constructions.

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arXiv2020

Coincidences between Calabi-Yau manifolds of Berglund-Hubsch type and Batyrev polytopes

In this article, we consider the phenomenon of complete coincidence of the key properties of pairs of Calabi-Yau manifolds realized as hypersurfaces in two different weighted projective spaces. More precisely, the first manifold in such a pair is realized as a hypersurface in a weighted projective space, and the second as a hypersurface in the orbifold of another weighted projective space. The two manifolds in each pair have the same Hodge numbers and special K\"ahler geometry on the complex structure moduli space and are associated with the same $N=2$ gauge linear sigma model. We give the explanation of this interesting coincidence using the Batyrev's correspondence between Calabi-Yau manifolds and the reflexive polyhedra.

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arXiv2019

GLSM for Calabi-Yau Manifolds of Berglund-Hubsch Type

In this note we briefly present the results of our computation of special K\"ahler geometry for polynomial deformations of Berglund-H\"ubsch type Calabi-Yau manifolds. We also build mirror symmetric Gauge Linear Sigma Model and check that its partition function computed by Supersymmetric localization coincides with exponent of the K\"ahler potential of the special metric.

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arXiv2019

Partition functions of $\mathcal{N}=(2,2)$ supersymmetric sigma models and Special geometry for the two-moduli non-Fermat Calabi-Yau manifold

We study the new case of the application of the JKLMR conjecture on the connection between the exact partition functions of $\mathcal{N}=(2,2)$ supersymmetric gauged linear sigma models (GLSM) on $S^2$ and special K\"ahler geometry on the moduli spaces of Calabi-Yau manifold $Y$. The last ones arise as manifolds of the supersymmetric vacua of the GLSM. We establish this correspondence using the Mirror symmetry in Batyrev's approach. Namely, starting from the two-moduli non-Fermat Calabi-Yau manifold $X$ we construct the dual GLSM with the supersymmetric vacua $Y$, which is the mirror for $X$. Knowing the special geometry on the complex moduli space of $X$ we verify the mirror version of the JKLMR conjecture by explicit computation.

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arXiv2018

JKLMR conjecture and Batyrev construction

We study a mirror interpretation of the relation between the exact partition functions of N=(2,2) gauged linear sigma-models (GLSM) on the 2d sphere and Kahler potentials on the moduli spaces of the CY manifolds proposed by Jockers et al. We use the Batyrev mirror construction for establishing the explicit relation between GLSM and the corresponding mirror family of the Calabi-Yau manifolds, defined as hypersurfaces in weighted projective spaces. We demonstrate how to do this by the explicit calculation in the case of the quintic threefold and its mirror.

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arXiv2018

Special geometry on Calabi--Yau moduli spaces and $Q$--invariant Milnor rings

The moduli spaces of Calabi--Yau (CY) manifolds are the special K\"ahler manifolds. The special K\"ahler geometry determines the low-energy effective theory which arises in Superstring theory after the compactification on a CY manifold. For the cases, where the CY manifold is given as a hypersurface in the weighted projective space, a new procedure for computing the K\"ahler potential of the moduli space has been proposed in \cite {AKBA1,AKBA2, AKBA3}. The method is based on the fact that the moduli space of CY manifolds is a marginal subspace of the Frobenius manifold which arises on the deformation space of the corresponding Landau--Ginzburg superpotential. I review this approach and demonstrate its efficiency by computing the Special geometry of the 101-dimensional moduli space of the quintic threefold around the orbifold point \cite {AKBA3}.

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arXiv2017

Special geometry on the 101 dimesional moduli space of the quintic threefold

A new method for explicit computation of the CY moduli space metric was proposed by the authors recently. The method makes use of the connection of the moduli space with a certain Frobenius algebra. Here we clarify this approach and demonstrate its efficiency by computing the Special geometry of the 101-dimensional moduli space of the quintic threefold around the orbifold point.

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arXiv2017

Special geometry on the moduli space for the two-moduli non-Fermat Calabi-Yau

We clarify the recently proposed method to compute a Special K\"ahler metric on a Calabi-Yau complex structures moduli space that uses the fact that the moduli space is a subspace of specific Frobenius manifold. We apply this method to computing the Special K\"ahler metric in a two-moduli non-Fermat model which has been unknown until now.

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